# COMPLEMENTARY AND SUPPLEMENTARY ANGLES

## About "Complementary and Supplementary angles"

"Complementary and supplementary angles"  is the much required stuff for the students who study math in school level.

Let us have a clear understanding about complementary angles and supplementary angles.

Complementary Angles  :

If the sum of two angles is 90⁰, then those two angles are called as complementary angles.

Example :

30° and 60° are complementary angles.

Because 30° + 60° = 90°.

Clearly, 30° is the complement of 60° and 60° is the complement of 30°.

Supplementary Angles  :

If the sum of two angles is 180⁰, then those two angles are called as supplementary angles.

Example :

120° and 60° are supplementary angles.

Because 120° + 60° = 180°.

Clearly, 120° is the supplement  of 60° and 60° is the supplement of 120°.

Let us see, how the stuff "complementary and supplementary angles" appears on picture.  ## Example problems

To have better understanding, let us do some problems on "Complementary and supplementary angles".

Example 1 :

The measure of an angle is 41°. What is the measure of a complementary angle?

Solution :

Let "x" be the measure of a complementary angle required.

Since "x" and 41° are complementary angles, we have

x + 41° = 90°

x = 90° - 41°

x = 49°

Hence the measure of the complementary angle is 49°

Example 2 :

The measure of an angle is 62°. What is the measure of a complementary angle?

Solution :

Let "x" be the measure of a complementary angle required.

Since "x" and 62° are complementary angles, we have

x + 62° = 90°

x = 90° - 62°

x = 28°

Hence the measure of the complementary angle is 28°

Example 3 :

The measure of an angle is 108°. What is the measure of a supplementary angle?

Solution :

Let "x" be the measure of a supplementary angle required.

Since "x" and 108° are supplementary angles, we have

x + 108° = 180°

x = 180° - 108°

x = 72°

Hence the measure of the supplementary angle is 72°

Example 4 :

The measure of an angle is 89°. What is the measure of a supplementary angle?

Solution :

Let "x" be the measure of a supplementary angle required.

Since "x" and 41° are supplementary angles, we have

x + 89° = 180°

x = 180° - 89°

x = 91°

Hence the measure of the supplementary angle is 91°

Example 5 :

Two angles are complementary. If one of the angles is double the other angle, find the two angles.

Solution :

Let "x" be one of the angles.

Then the other angle = "2x"

Since "x" and "2x" are complementary angles, we have

x + 2x = 90°

3x = 90°

x = 30° and 2x = 60°

Hence the two angles are 30° and 60°

Example 6 :

Two angles are complementary. If one angle is two times the sum of other angle and 3, find the two angles.

Solution :

Let "x" and "y" be the two angles which are complementary.

So, we have x + y = 90° --------> (1)

From the information, "one angle is two times the sum of other angle and 3", we have

x = 2(y+3)

x = 2y + 6 ------->(2)

Now plug x = 2y + 6 in equation (2)

(1)-------> 2y + 6 + y  = 90

3y + 6 = 90

3y = 84

y = 28

Now, plug y = 28 in equation (2).

(2) --------> x = 2(28) + 6

x = 56 + 6

x = 62

Hence the two angles are 62° and 28°

Example 7 :

The measure of an angle is 3/4 of 60°. What is the measure of the complementary angle ?

Solution :

Let "x" be the measure of a complementary angle required.

Given angle = 3/4 of 60° = (3/4)x60° = 3x15° = 45°

Since "x" and 45° are complementary angles, we have

x + 45° = 90°

x = 90° - 45°

x = 45°

Hence the measure of the complementary angle is 45°

Example 8 :

Two angles are supplementary. If one angle is 36° less than twice of the other angle, find the two angles.

Solution :

Let "x" and "y" be the two angles which are supplementary.

So, we have  x + y = 180° ----------->(1)

From the information, "one angle is 36

From the information, "one angle is 36° less than twice of the other angle", we have

x = 2y - 36 ----------->(2)

Now plug x = 2y - 36 in equation (1)

(1)-------> 2y - 36 + y  = 180

3y - 36 = 180

3y = 216

y = 72

Now, plug y = 72 in equation (2).

(2) --------> x = 2(72) - 36

x = 144 - 36

x = 108

Hence the two angles are 108° and 72°

Example 9 :

An angle and its one-half are complementary. Find the angle.

Solution :

Let "x" be the required angle.

Its one half is x/2

Since "x" and "x/2" are complementary, we have

x + x/2 = 90°

(2x + x)/2 = 90°

3x/2 = 90°

3x = 180°

x = 60°

Hence the required angle is 60°

Example 10 :

Two angles are supplementary. If 5 times of one angle is 10 times of the other angle. Find the two angles.

Solution :

Let "x" and "y" be the two angles which are supplementary.

So, we have x + y = 180° -------->(1)

From the information, "5 times of one angle is 10 times of the other angle", we have

5x = 10y  =====> x = 2y ---------(2)

Plug x = 2y in equation (1)

2y + y = 180

3y = 180

y = 60

Plug y = 60 in equation (2)

x = 2(60)

x = 120

Hence the two angles are 60° and 120°.

So far we have seen problems on "complementary and supplementary angles  "without pictures

Apart from the above examples, let us do some problems on "Complementary and supplementary angles" with pictures.

Example 11 :

Find the value of  "x" in the figure given below. Solution :

From the picture above, it is very clear that the angles "x" and "2x" are complementary.

So, we have x + 2x = 90°

3x = 90°

x = 30°

Hence the value of "x" is 30°

Example 12 :

Find the value of  "x" in the figure given below. Solution :

From the picture above, it is very clear that the angles (x+1), (x-1) and (x+3) are complementary.

So, we have (x+1) + (x-1) + (x+3) = 90

3x + 3 = 90

3x = 87

x = 29

Hence the value of "x" is 29

Example 13 :

Find the value of  "x" in the figure given below. Solution :

From the picture above, it is very clear that (2x+3) and (x-6) are  supplementary angles.

So, we have (2x+3) + (x-6)  = 180°

2x + 3 + x - 6 = 180°

3x - 3 = 180

3x = 183

x = 61

Hence the value of "x" is 61.

Example 14 :

Find the value of  "x" in the figure given below. Solution :

From the picture above, it is very clear (5x+4), (x-2) and (3x+7) are supplementary angles.

So, we have (5x+4) + (x-2) + (3x+7) = 180°

5x + 4 + x -2 + 3x + 7 = 180°

9x + 9 = 180

9x = 171

x = 19

Hence the value of "x" is 19

After having gone through the  stuff and examples on "Complementary and supplementary angles", we hope that the students would have understood the stuff "Complementary and supplementary angles".

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